Financial aid refund exceeding tuition

Why can't you be claimed by anyone else? 

Yes, there is a loop hole available. See below 2nd line for details.  Most college age students are not eligible for the refundable credit.  See below 1st line for details. 

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There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, even if you don't qualify as a dependent, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. It is usually best if the parent claims that credit. 

You cannot claim the (up to) $1000 refundable credit if you are, or can be, claimed as a dependent by someone else.

Reference: Line 7 instructions for form 8863.

https://www.irs.gov/instructions/i8863#en_US_2024_publink53002gd0e674

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There is a tax “loop hole” available to claim an education credit, for the parents of students on scholarship. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents  (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship.  You cannot do this  if the conditions of the grant are that it be used to pay for qualified expenses (Pell grants are not restricted like that).

Using your numbers: Student has $6002 in box 5 of the 1098-T and $4646 in box 1. At first glance he/she has $1356 of taxable income and nobody can claim the American opportunity credit. But if she reports $5356 as income on her return, the parents can claim $4000 of qualified expenses on their return.

Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket. She would only need to report $4356 of taxable scholarship income, instead of $5356.

The IRS actually encourages use of this technique. From the form 1040 instructions: “You may be able to increase an education credit if the student chooses to include all or part of a Pell grant or certain other scholarships or fellowships in income. For more information, see Pub. 970, the instructions for Form 1040 and IRS.gov/EdCredit".  PUB 970 even has examples of how to do the “loop hole”.